Jim and Katharine Miller Diehl thought they had found their ticket out of the United States: a three-bedroom condo in a new development overlooking the Mediterranean coast in southern Spain. The retired couple from St. Petersburg, Fla., were banking on Spain’s popular residency-by-investment program, known as the “golden visa.”

“Living abroad has been a lifetime dream,” said Ms. Miller Diehl, 63. “We just were attracted to Spain. The cost of living is a lot less than in the States. The pace of life is slower. The food is great.”

That was three years ago. They never anticipated it would be a race against the clock. In March, with construction of their condo nearing completion, the couple hurried to Spain to formally apply for the program, which granted residency rights to foreigners who bought property worth at least 500,000 euros. But now they were staring down a fast-approaching deadline: April 3.

“It wasn’t supposed to be a time bomb,” Mrs. Miller Diehl said. “I had to go dye my hair last week. I’ve grown a bunch of gray hairs.”

Last year, the Spanish government abruptly announced that it would stop issuing visas by investment, part of a new approach to combat a housing crisis that had pushed rents and home prices out of reach for many native Spaniards. For foreign home buyers, the door to a new life was closing.